# prime number theory1

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IntroductionArithmetic progressions

Other linear patterns

Recent progress in additive prime numbertheory

Terence Tao

University of California, Los Angeles

Mahler Lecture Series

Terence Tao Recent progress in additive prime number theory

IntroductionArithmetic progressions

Other linear patterns

Additive prime number theory

Additive prime number theory is the study of additivepatterns in the prime numbers 2,3,5,7, . . ..Examples of additive patterns include twins p,p + 2,arithmetic progressions a,a + r , . . . ,a + (k 1)r , andprime gaps pn+1 pn.Many open problems regarding these patterns still remain,but there has been some recent progress in somedirections.

Terence Tao Recent progress in additive prime number theory

IntroductionArithmetic progressions

Other linear patterns

Random models for the primesSieve theorySzemerdis theoremPutting it together

Long arithmetic progressions in the primes

Ill first discuss a theorem of Ben Green and myself from2004:Theorem: The primes contain arbitrarily long arithmeticprogressions.

Terence Tao Recent progress in additive prime number theory

IntroductionArithmetic progressions

Other linear patterns

Random models for the primesSieve theorySzemerdis theoremPutting it together

It was previously established by van der Corput (1929) thatthe primes contained infinitely many progressions of lengththree. In 1981, Heath-Brown showed that there areinfinitely many progressions of length four, in which threeelements are prime and the fourth is an almost prime (theproduct of at most two primes).The proof of the full theorem combines three separateingredients: random models for the primes, sieve theory,and Szemerdis theorem.

Terence Tao Recent progress in additive prime number theory

IntroductionArithmetic progressions

Other linear patterns

Random models for the primesSieve theorySzemerdis theoremPutting it together

Prime counting heuristics

While we are not able to prove everything we would like toin this subject, we do have a rather convincing set ofheuristics with which to predict how to count variouspatterns in the primes.The starting point is the prime number theorem, whichasserts that the number of primes less than a largenumber N is roughly N/ logN.One can interpret this fact probabilistically: if one picks aninteger at random between 1 and N, then it has aprobability of about 1/ logN of being prime.

Terence Tao Recent progress in additive prime number theory

IntroductionArithmetic progressions

Other linear patterns

Random models for the primesSieve theorySzemerdis theoremPutting it together

Cramrs random model

Cramrs random model for the primes asserts that theprimes behave as if each integer n had an independentprobability of 1/ logn of being prime, in the sense thatstatistics for the primes asymptotically match statistics forthis random model.This model turns out to not be totally accurate, but thereare some refinements to this model which give quiteconvincing predictions.Well illustrate this with a study of the twin primeconjecture: there are infinitely many pairs p,p + 2 ofprimes that are a distance 2 apart. This ancient conjectureremains open, despite many partial results.

Terence Tao Recent progress in additive prime number theory

IntroductionArithmetic progressions

Other linear patterns

Random models for the primesSieve theorySzemerdis theoremPutting it together

A proof of the twin prime conjecture

Let N be a large number, and let n be an integer chosenrandomly between 1 and N.From the prime number theorem, n and n + 2 each have aprobability about 1/ logN of being prime.Assuming that the events n is prime and n + 2 is primeare independent, we conclude that n,n + 2 aresimultaneously prime with probability about 1/ log2 N.In other words, there are about N/ log2 N twin primes lessthan N. Letting N we obtain the twin primeconjecture.

Terence Tao Recent progress in additive prime number theory

IntroductionArithmetic progressions

Other linear patterns

Random models for the primesSieve theorySzemerdis theoremPutting it together

Correcting the model

This argument is incorrect. One reason to see this is that itwould also predict an infinite number of consecutive primesn,n+ 1, which is false, as all but one of the primes are odd.However, one can correct for this by refining the model.Right now, we are giving each integer n {1, . . . ,N} anequal chance of 1/ logN of being prime. A smarter modelwould be to give the odd integers a 2/ logN chance ofbeing prime and the even integers a 0 chance of beingprime. (This omits the prime 2, but this is negligible in thegrand scheme of things.)With this refined model, consecutive primes are ruled out(as they should), and the expected number of twin primesincreases from N/ log2 N to 2N/ log2 N.

Terence Tao Recent progress in additive prime number theory

IntroductionArithmetic progressions

Other linear patterns

Random models for the primesSieve theorySzemerdis theoremPutting it together

The prime tuples conjecture

One can refine the model further, by excluding themultiples of 3 from being prime (and increasing theprobability of the remaining numbers of being prime from2/ logN to 3/ logN). This turns out to adjust the expectednumber of twin primes downward, to 1.5 N

log2 N.

Continuing to add information about small moduli, theexpected count given by these models continues tochange, but can be easily computed to converge to anasymptotic prediction, which in the case of twin primesturns out to be 2 Nlog2 N , where 2 is the twin primeconstant

2 = 2

p odd prime(1 1

(p 1)2 ) 1.320 . . . .

Terence Tao Recent progress in additive prime number theory

IntroductionArithmetic progressions

Other linear patterns

Random models for the primesSieve theorySzemerdis theoremPutting it together

This is believed to be the correct asymptotic.More generally, there is a similar asymptotic conjecturedfor other patterns in the primes; this is basically theHardy-Littlewood prime tuples conjecture. Roughlyspeaking, it is asserting that the sequence of adjustedCrmer models discussed earlier is asymptoticallyaccurate for describing the primes.

Terence Tao Recent progress in additive prime number theory

IntroductionArithmetic progressions

Other linear patterns

Random models for the primesSieve theorySzemerdis theoremPutting it together

One can think of each of these models as identifying acertain amount of structure in the primes, and thensaying that all other aspects of the primes are random.For instance, one could observe the structure that theprimes have density about 2/ logN in the odd numbersand 1/ logN in the even numbers, but assert that there isno discernible additional structure on top of this.

Terence Tao Recent progress in additive prime number theory

IntroductionArithmetic progressions

Other linear patterns

Random models for the primesSieve theorySzemerdis theoremPutting it together

Viewed in this light, the prime tuples conjecture isasserting that apart from the obvious structure that theprimes obey (they are almost all coprime to 2, coprime to3, etc.), there is no additional pattern or structure to thissequence of integers, and they behave as if they wererandom relative to the structure already identified.However, we are unable at this time to rigorously rule out abizarre conspiracy among primes to exhibit an additionallayer of structure (e.g. to avoid congregating as twinsn,n + 2 after a certain point). How does one disprove aconspiracy?

Terence Tao Recent progress in additive prime number theory

IntroductionArithmetic progressions

Other linear patterns

Random models for the primesSieve theorySzemerdis theoremPutting it together

Sieve theory

Now we turn from random models to another aspect ofprime number theory, namely sieve theory.One way to approach the primes is to start with all theintegers in a given range (e.g. from N/2 to N) and then siftout all the non-primes, for instance by removing themultiples of 2, then the multiples of 3, and so forth up tothe multiples of

N (the sieve of Eratosthenes).

One can hope to count, say, twin primes, by keeping trackof the number of twins at each stage of the sifting process.

Terence Tao Recent progress in additive prime number theory

IntroductionArithmetic progressions

Other linear patterns

Random models for the primesSieve theorySzemerdis theoremPutting it together

For instance, the number of twins n,n + 2 in the entirerange [N/2,N] is N/2 + O(1). After removing the multiplesof two, the count drops to N/4 + O(1); after removing themultiples of three, it drops further to N/12 + O(1), and soforth.Unfortunately, the O(1) errors multiple rapidly, andoverwhelm the main term long before one reaches thelevel of multiples of

N.

One can partially address this problem by smoothing thesieve (rather than eliminating numbers outright, adjust theirscore upward or downward whenever they are divisible ornot divisible by certain numbers), but one still cannot get toN by these techniques alone (there is a specific

obstruction to this, known as the parity problem).

Terence Tao Recent progress in additive prime number theory

IntroductionArithmetic progressions

Other linear patterns

Random models for the primesSieve theorySzemerdis theoremPutting it together

Almost primes

Nevertheless, it is possible to use sieve theory to controlthings up to some partial